group decision-making: head-count versus intensity of preference

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Socio-Economic Planning Sciences 41 (2007) 22–37

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Group decision-making: Head-count versusintensity of preference

Thomas L. Saaty�, Jen S. Shang

The Katz Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260, USA

Available online 1 December 2005

Abstract

This paper puts forth a framework for reshaping the group decision-making process. The proposed framework extends

from the usual one-issue-at-a-time decision-making to one that involves several related issues simultaneously. Weaknesses

of the traditional majority voting mechanism are first identified, and then a different voting method that takes each

individual voter’s sentiment into account is discussed. Specifically, a decision-maker is asked to express his/her intensity of

preference for the issues encountered. Three hierarchical structures—benefits, costs, and risks—are developed to evaluate

the alternatives. Due to the nature of pairwise comparisons and synthesis, the proposed method is amenable to consensus

building and has higher reliability and consistency. It can be used for candidate selection, e.g. governmental election, when

a large population is involved. It is also effective for resource allocation and prioritization when a small group or business

is concerned. We believe the proposed approach has potential for resolving deficiencies of the conventional voting

mechanism, and can be applied to many real-world problems. Its implementation on the Internet is also discussed.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Group decision-making; AHP; Multiple-issue agendas; Voting; Intensity of preference; Internet

1. Introduction

Many researchers have studied the problem of combining individual preferences to form a consensus ofopinion or compromise. Problems of this nature often appear in resource allocation, project selection, andpolicy-making. Conventional wisdom regarding public policy-making is grounded in the widespread majorityvote mechanism. That is, either a simple or a two-thirds majority vote determines the final decision, and theminority must unconditionally compromise its position. It is a winner-take-all outcome. The losers’ possiblestrong preferences for the opposite alternative are no longer important, and their cooperation with, anddeference to the will of, the majority are expected. While convenient, the current voting system oversimplifiesthe representation of voter preferences and ‘‘drowns out’’ the true merit of counterarguments. In spite of itsfairness, in principle and often in practice, opposing opinions are ignored, and this may be painful to thelosers. We wonder if this approach to democracy is ordained by divinity, generated through our biology, orimprovised by human rationality. Much attention has been paid by the utility theory researchers to study

e front matter r 2005 Elsevier Ltd. All rights reserved.

ps.2005.10.001

ing author.

esses: [email protected] (T.L. Saaty), [email protected] (J.S. Shang).

www.elsevier.com/locate/seps

ARTICLE IN PRESST.L. Saaty, J.S. Shang / Socio-Economic Planning Sciences 41 (2007) 22–37 23

group decision-making problems (see [1–4]). However, the absence of a formal, dominating, and widelyaccepted theory to aggregate cardinal preferences may be a stumbling block that prohibits us from movingbeyond the traditional ordinal approach.

1.1. Literature review

Using ballots to solicit the inclination of individuals in a group has been a subject of great interest for nearly200 years. In Group Choice, Mirkin [5] scanned the diverse horizons of the field. He found that much of theresearch had focused on the ordinal representation of preferences, and on the problems and pitfalls of theordinal approach. Barbut [6] constructed examples to illustrate paradoxes of the ordinal approach whenvoting on three alternatives. Several cases were subsequently shown and demonstrated to be contradictory.This eventually led to the development of the well-known Arrow [7] impossibility theorem for ordinalpreferences. The theorem states that if the number of alternatives is greater than two, it is impossible to createa group preference ordering that satisfies four seemingly natural conditions that one would expect to hold.These are non-dictatorship, decisiveness, Pareto optimality (agreement), and independence of irrelevant

alternatives.To remove the contradiction outlined by Arrow, three types of ordinal methods were attempted in the

literature: preference scoring, distance-based methods, and statistical methods. These are intended to relax oneor the other of the four conditions and, in particular, the fourth one. However, the results are unsatisfactory,at least for addressing the question of the general uniqueness of the outcome regardless of the method used.

In addition to the work of Armstrong et al. [8] and Cook and Seiford [9], Cook and Kress [10] developed amodel for aggregating ordinal rankings to express intensity of preference. Mueller [11] and Plott [12] providedan extensive survey for consensus ranking through generalized network formulation. The debate hascontinued because the roots of impossibility lie in the use of ordinal preferences.

It was once thought that a cardinal approach to aggregating individual preferences is not plausible.MacKay [13] thus writes that pursuing the cardinal approaches is like chasing what cannot be caught.Nevertheless, by considering problems in arms control negotiations, Saaty [14,15] developed a general theoryof measurement based on absolute scales called the Analytic Hierarchy Process (AHP) that does precisely that.AHP provides a method for aggregating individual cardinal preferences into a unique group preference whileremoving impossibility, as shown by Saaty and Vargas [16]. Because it deals with measurement, AHPfacilitates the group process to capture preference intensities of individuals and incorporates them into a finalgroup decision. It ensures the validity of the outcome as it relates to the real world, a question rarely addressedin the ordinal approach.

In the current paper, we illustrate the use of the cardinal approach. It is organized as follows. In Section 2,we discuss the deficiency of the traditional decision-making (voting) system; Section 3 provides the frameworkfor applying the AHP to voting, and for analyzing the sensitivity of the proposed method. In Section 4, wediscuss its implementation on the Internet. Summary and conclusions are made in Section 5.

2. Deficiencies of the traditional yes–no voting system

The traditional voting method requires voters to choose between ‘‘yes’’ and ‘‘no’’ for an alternative. Manyregard this (1–0) majority voting method as an unchallengeable law of nature. It is because, thus far, we havenot found a way of voting that is more practical and better represents the decision-makers’ true preferences. Inthis section, we examine the deficiencies of the traditional (1–0) head-count procedure.

First, with a majority vote, individuals are unable to express their true preference for the subject of a debatewithout eventually taking the most extreme position by either voting for it or against it. A person may preferone issue over its opposite only by a proportion of 51 to 49 percent. Yet, when that person votes, the vote isrecorded as definitely for ( ¼ 1) or definitely against ( ¼ 0). If many people vote with lukewarm feelings, theoutcome indicates a stronger win than is justified by reality. Decision-making under such circumstances issubject to extreme bias.

Second, when the issues involved are of public concern, it may not be appropriate to resolve them throughthe familiar process of competitive voting. The danger of basing decisions on head-count is that the win/lose

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Table 1

Head-count method vs. intensity rating of preference

Issue # Number of voters for ‘‘Yes’’ or ‘‘No’’ Geometric mean based on the

preference intensity of the voters

Majority

outcome

Outcome based

on preference

intensity

Yes No Yes No

1 19 21 0.55 0.45 No Yes

2 10 30 0.38 0.62 No No

3 23 17 0.47 0.53 Yes No

4 22 18 0.61 0.39 Yes Yes

T.L. Saaty, J.S. Shang / Socio-Economic Planning Sciences 41 (2007) 22–3724

dynamic may not be good for cases where success depends on cooperation and teamwork. The winner-take-allmethod may be appropriate for a society facing a war and seeking to win, but, perhaps, is not as suitable whencollaborative effort is essential for getting along.

A third flaw of the yes–no voting system is that the decision derived from a majority vote may result in anoutcome that is the opposite of what the collectivity wants. For example, suppose there are three people votingon two alternatives, A and B. Two people have intensities of preference of 45% for A, and 55% for B. Thethird person has preference intensity of 90% for A, and 10% for B. Under the yes–no voting system, B wins bya simple majority 2:1 vote. If the intensities of preference are taken into account, the mean preference intensityfor A and B (using either of the two well-known and mathematically advocated methods in the literature, thearithmetic and geometric means) would be (0.45+0.45+0.90)/3 ¼ 0.60 and (0.55+0.55+0.10)/3 ¼ 0.40;and (0.45� 0.45� 0.90)1/3 ¼ 57% and (0.55� 0.55� 0.10)1/3 ¼ 31%, respectively. Consequently, A wins over B,exactly the opposite of the yes–no approach.

A fourth potential difficulty arises in voting on several issues at the same time (agenda effects). Whenmultiple issues are encountered, the traditional voting approach takes on each issue separately. If the issues arebound together (dependent) to some extent, it can happen that an earlier issue with bearing on what followsis voted out, killing the chance to successfully influence the others, unless it is brought back again forreconsideration.

Such yes–no voting often prevents the decision-makers from following a comprehensive view of issues as awhole. It can then lead to a chain of policies that is hard to carry out, or, at best, makes it less efficient tocreate what the public sentiment is asking for. This opens the floodgate for unsettling and paradoxical results.A better approach might be to discuss all the relevant issues simultaneously and make decisions on them witha ranking of the issues.

The difference between the yes–no head-count method and that of using intensity of preference in decision-making is illustrated in Table 1. The example includes 40 voters. Four issues are listed in the first column. Thesecond and third columns list the number of people voting in favor of or against each issue. Each of the votersis also asked to express his/her intensity of preference on both the ‘‘yes’’ and ‘‘no’’ alternatives. The geometricmeans of all 40 voters’ for and against the issues are listed in the fourth and fifth columns. The sixth columngives the outcome of the decision on each issue from the majority vote in the second and third columns, whilethe seventh column gives the outcome from the preference intensities in the fourth and fifth columns.

The example demonstrates that when various issues are deliberated concurrently, each possible outcome (yesor no alternative) under an issue has different chances of success. The voter must specify intensities ofpreference for all issues and alternatives and form an ordered set of preferences. Under such circumstances, onecannot exaggerate the importance of a specific alternative without making the others suffer. Since more thanjust a 1 or a 0 is required, a decision-maker is forced to think more about the strength of preference (s)he isasked to provide. Thus, decision-making becomes more substantive and less of a ‘‘muddling through’’ process.

3. Applying the AHP to voting

Policy-making requires inputs of eligible individuals or representatives. Our example in this section focuseson the public policy-making issues encountered by a legislative body. The political example in Section 3.1 is


meant to give a slightly more serious flavor to the analysis and to draw the attention of readers to mattersabout which the public is usually concerned. Our hope is to invite a broader and strategic look at ourapproach. The views expressed here are drawn from newspapers, magazines, and Internet articles. They formthe basis for constructing the hierarchies and judgments in Section 3.2. The variables used in the model, andthe values assigned to them, are purely illustrative. Nevertheless, it would not diminish the importance of theidea we are presenting in this paper. In the following, we discuss the legislative matters that are important tothe US public. Those viewpoints serve as foundation for walking the readers through the proposed newmethod.

3.1. Current events

In today’s society, the majority wishes to lower taxes, eradicate discrimination, and reform the politicalsystem. When properly taken into account, these concerns would lead to new policies essential for reshapingsociety. If mismanaged, they can increase racism, intensify economic class war, and drag the nation intodissatisfaction, negativism and recession. The challenge is to transform the power of national sentiment intorealistic, constructive policies that make the government more productive and less costly. To illustrate thesituation faced by our AHP model, we focus on the following issues.

3.1.1. Tax reform

To increase the nation’s supply of capital, which, in turn, may lead to more jobs and higher living standards,and lay the groundwork for growth and prosperity in the long run, the Bush administration proposed toeliminate the tax on dividends, accelerate income tax cuts, offer investment incentives for small businesses, andprovide funds for job re-training [17]. While it makes sense to eliminate the double taxation of dividends, newtax cuts indeed may embroil the nation in deficit and create higher interest rates that cut growth. Critics alsoargue that tax reform skews the benefits to wealthier taxpayers and that it will not help the middle class,stimulate investment spending, or spur growth. Hence the benefits may be few.

3.1.2. Affirmative action reform

The US Constitution demands that all citizens be treated equally. Yet, racial discrimination remains a factof our society [18]. The question is: should affirmative action continue to exist, or should it be unplugged?Opponents argue that the current policy of employment and college admission based on an applicant’s colorof skin is discriminatory [19]. Supporters believe that affirmative action is necessary as long as racialdiscrimination persists, and that our nation can function better when diverse members of the population co-exist harmoniously [20]. Thus, some have argued that we should balance short-term pressures with long-termsustainability, and continue practicing affirmative action [20].

3.1.3. Legislative term limits

Many regard term limits as a way to end the unlimited control of a few elites and to create a fairer politicalsystem [21]. Opponents argue that limiting terms restricts citizens’ rights to choose, and treats competent orincompetent representatives the same [22]. Further, critics argue that term limits may cause representatives tooverlook the needs of the people they represent, since seeking long-term support from their districts is not aconcern [22]. Moreover, the goal of eliminating the oftentimes cozy relationship between legislators andspecial interest groups may not be realized by implementing term limits alone [22]. Some argue that a betteridea would be strong campaign finance reform, which would deprive incumbents of their considerablefinancial advantage [21].

3.2. Applying AHP to policy-making

The political opinions assembled in Section 3.1 provide the necessary foundation for readers to understandthe AHP example to be illustrated in this section. Here, we give details of how the AHP can be used toimprove the policy-making process. To give a comprehensive view, we divide the above factors that influencedecisions into three hierarchies: one for the benefits of implementing certain policies, one for the costs, and

ARTICLE IN PRESST.L. Saaty, J.S. Shang / Socio-Economic Planning Sciences 41 (2007) 22–3726

a third for the risks and uncertainties that can arise. Each hierarchy has a goal followed by the criteria thataffect the performance of that goal. The issues are listed at the bottom level of the hierarchy. Our purpose hereis to walk the reader through practical examples, both to improve his/her understanding of the main idea ofthe paper, and to show how it can be implemented in real-world decision-making.

3.2.1. Implementation in a group

In AHP, a nine-point scale of absolute numbers is adopted as the unit for comparison and used torepresent participants’ judgments as to importance, likelihood, or preference among homogeneousoptions. The scale expresses how many times more one of two options is preferred to the other. Oddnumbers from 1 to 9 are used to represent the pairwise comparison judgments from equal to extremepreference, with 1, 3, 5, 7, and 9 standing for equal, moderate, strong, very strong, or extremely strongpreference. Even numbers in the intervals are used to present intermediate values for which there is no conciseverbal expression. The reciprocal value represents preference for the less preferred option over the morepreferred one.

When a group such as a legislative body is making a decision, it is necessary to aggregate the preferences ofindividuals into a consensus rating. There are two ways to combine the numerical pairwise comparisonjudgments of individuals to form a judgment for the group. The first is consensus vote. This requires the groupto reach an agreement on the value of each entry in a matrix of pairwise comparisons, but this is usually hardto attain.

A better way is to take the geometric mean. The basis for using the geometric rather than the arithmeticmean to combine judgments of different individuals has been justified mathematically first by Saaty [15, p. 32],and later by Aczel and Saaty [23]. They proved that with the conditions of separability, associativity,cancellativity, consensus, and homogeneity needed to synthesize judgments of individuals, one could use thearithmetic or geometric means. However, when reciprocity is used, the geometric mean is the only way tocombine judgments. Aczel and Alsina [24] proved that if the voters have different power of influence withcorresponding priority for each, then the judgment of each individual is raised to that priority (priority0owio1,

Pwi ¼ 1) first, and then taking their product. Along these lines if we have M groups of different

sizes, with ni members in group i, in which the individuals arrive at a consensus by using the geometric meanwithin each group, then the final outcome for all the groups is obtained by raising each group’s geometricmean to the power of ni=

PMi¼1ni, and then taking their product. Here each group becomes like an individual

with the number in the group serving as that individual’s relative power.Given N decision-makers, the elements aij of the preference matrix aggregated by the geometric

mean is: aij ¼ ½QN

k¼1aKij �

1=N , where akij is the judgment of the kth voter when comparing item i with item j.

For example, if four committee members regard the relative importance of the tax reform benefits overaffirmative action reform benefits as 4, 6, 5, and 3, respectively, then the aggregate importance of tax reformwould be (4� 6� 5� 3)1/4 ¼ 4.36. That is, individual judgments are replaced by the geometric mean forthe group.1

3.2.2. The analysis framework

Fig. 1 presents a framework for problems dealing with multiple issues and various criteria simultaneously.When applying the model to the national issues discussed above, a decision-maker first assesses the relativeimportances of the issues under the benefits, costs and risks hierarchies. The analysis would be followed by adecision as to whether or not reform is needed. This is done by comparing the two alternatives—to reform ornot to reform—with each other under a criterion. Proportionality of the rankings—the three issues amongthemselves and the two alternatives (reform, no reform) within each issue—makes it possible to integrate allsix alternatives into a single rank order under a hierarchy.

The importance (priority) of the issues indicates the relative commitment with which they would each becarried out if they must all be implemented. Our task is to determine which ones may or may not beimplemented in an overall consideration of benefits, costs, and risks relative to all the issues involved and not

1It is easy to see that the reciprocal of this value (i.e., 1/(4.36) ¼ 0.229) is the same as that obtained by applying the geometric mean to

the reciprocals, i.e. (1/4� 1/6� 1/5� 1/3)1/4 ¼ 0.229.

ARTICLE IN PRESS

Step 1 - Assess the relative importance of each issue

(a) Compute cij = the weight of criterion j under hierarchy i. - pairwise compare the importance of each criterion under each hierarchy (see Figure 3). (b) Compute wijk = the relative importance of issue k contributing to criterion j under hierarchy i. - pairwise compare the importance of each issue under each criterion (see Figure 4).

j 1Σ=

J

ijc = 1 (for i=1, 2,...I. I is the total number of hierarchies; J=the total number of criteria under

hierarchy i)

Σ=

K

kijkw

1= 1 (K=the total number of issues) (see Figures 2a-c)

Step 2 – Determine the importance of each alternative

See Figure 5. (a) Compute the local priority: pijk

l

- pairwise compare the two alternatives (l=1, 2) of issue k, under criterion j in hierarchy i.

(b) Derive the global priority: gijkl = pijk

l × wijk

- multiply the local priority of each alternative with its corresponding issue’s weight, wijk .

(c) Derive the Local Rating: LRikl = l

ijk

J

jij pc ×Σ (for i=1,2...I; k=1,2,…K; l=1,2)

- sum the product of the alternative’s local priority with its corresponding criteria’s weight, cij .

(d) Derive the Overall Rating: ORikl

= lijk

J

jij gc ×Σ (for i=1,2...I; k=1,2,…K; l=1,2)

- sum the product of the alternative’s global priority with its corresponding criteria’s weight, cij .

Step 3 - Derive the ratio

For each alternative, use the Overall Rating from each hierarchy and derive the ratio, R:

Benefits

Costs Risks× = Rk

l =

lk

lk

lk

OROR

OR

,3,2

,1

×for k=1,2,3 and l=1,2

Step 4 – Make decision

Choose that alternative with the highest ratio within each issue k

Dk =arg maxl { Rkl} for k=1,2,3

Fig. 1. The AHP voting procedure for dealing with various issues and multiple alternatives.

T.L. Saaty, J.S. Shang / Socio-Economic Planning Sciences 41 (2007) 22–37 27


simply in isolation. Of course, for each pair to change, or to preserve the status quo, only one would bechosen. But, the one chosen is then determined as a function of the overall priorities. Details are given below.

(i)
Determining the importance of each issue, wijk.Figs. 2a–c display the benefits, costs, and risks hierarchies needed to assess the importance of the three
reform issues. The goal is at the top of each hierarchy, followed by the criteria that contribute to attainingthat goal. At the bottom of each hierarchy are the issues whose priorities are to be determined. Theweights, cij , for the criteria in the second level of each hierarchy are derived by pairwise comparisons andsynthesized as illustrated in the matrices of Fig. 3a. Each matrix specifies the judgments of the decision-maker about the relative importance of each criterion in terms of its contribution to the achievement ofthe goal of that hierarchy.

For example, in the benefits hierarchy, a possible question is: How much more important is promotingharmony in society over the importance of stimulating employment? Assume that social harmony isstrongly believed to be more of a serious concern vs. stimulating employment; a priority value of 5 is thusassigned. When a group is involved, each individual needs to provide his/her own judgment; members’final judgments are then combined by taking the geometric mean.Let A be an n� n preference matrix constructed through pairwise comparisons of n ‘‘objects.’’ Thepriorities derived from a pairwise comparison reciprocal matrix of judgments is obtained by solving

Xn

j¼1

aijwj ¼ lmax wi, (1)

Xn

i¼1

wi ¼ 1Xn

i¼1

wi ¼ 1 (2)

with aji ¼ 1=aij, aij40 (A is known as a reciprocal matrix) whose solution, known as the principaleigenvector, is normalized as in (2). For simplicity, Fig. 3b gives an approximation procedure forgenerating the priority for the benefits hierarchy.The third level of the hierarchy in Fig. 2a shows the importance of each issue, wijk’s, i.e, how it contributesto each criterion. Its pairwise comparisons are detailed in Fig. 4. As an example, in the employmentmatrix in the benefits hierarchy, the tax reform issue is considered to be very strongly more important incontributing to increasing employment over affirmative action reform. The former is thus assigned thevalue 7 when compared with the latter. The three issues are compared with respect to their contributionsto each criterion. Relative importance, or priority, wijk, is listed in the right-hand column of each matrix.Next, we copy all derived scales into Figs. 2a–c and weight the importance of an issue by thecorresponding cij to obtain the overall importance of that issue (i.e., the rightmost column). For example,the importance of tax reform in the benefits hierarchy is 0.682, using

ð0:777� 0:110Þ þ ð0:773� 0:238Þ þ ð0:687� 0:164Þ þ ð0:614� 0:488Þ ¼ 0:682.

Note that the benefits hierarchy (Fig. 2a) determines which issue yields the greatest benefits with respectto each criterion. The costs and risks hierarchies (Figs. 2b and c), respectively, decide the relative costs andrisks of each issue. From those, we find that tax reform is likely to generate the most benefits, whileaffirmative action reform is most costly and has the highest risk.

(ii)
Determining the local and overall rating for each alternative, LRlik and ORl
ik

This step distinguishes our model from the traditional AHP application. Having developed the issues’priorities in the respective hierarchies, we return to the third level of each hierarchy in Fig. 2. The threeissues are then replaced by three pairs of alternatives, one pair for the actions of the tax reform issue,another for the actions of the affirmative action issue, and the third for the term limit issue. Each pairrepresents (i) the status quo, and (ii) the potentially new state obtained by changing from the statusquo (see Fig. 5). A tax reform rating of 0.634 ¼ (0.875� 0.110)+(0.889� 0.238)+(0.800� 0.164)+(0.400� 0.488) and a no tax reform rating of 0.366 can be derived through step 2(c) of Fig. 1. Thecorresponding column in Fig. 5a is named ‘‘Local Rating’’ (LR) since only one issue is considered at

ARTICLE IN PRESS

IncreaseEmployment

0.110

Stimulate Investment &

Economy0.238

StrengthenTechnology &Infrastructure

0.164

PromoteSocietal

Harmony0.488

cij

ImportanceTax Reform Issue 0.777 0.773 0.687 0.614 0.682Affirmative Action

Reform Issue 0.153 0.134 0.127 0.117 0.127 Term Limit Issue 0.070 0.093 0.186 0.268 0.191

wijk

Benefits

Deepened Deficit

0.143

Unfairness Perception

0.452

Less Effective Government

0.212

Costs

FewerOpportunities

& Choices0.193

ImportanceTax Reform Issue 0.806 0.236 0.149 0.122 0.277 Affirmative Action

Reform Issue 0.117 0.682 0.160 0.320 0.421 Term Limit Issue 0.077 0.082 0.691 0.558 0.302

Long-TermNegative

Competitiveness 0.540

Class War

0.297

Unfair Burden on Future

Generation 0.163

Risks

ImportanceTax Reform Issue 0.260 0.300 0.783 0.357Affirmative Action

Reform Issue 0.413 0.600 0.155 0.426Term Limit Issue 0.327 0.100 0.062 0.217

(a)

(b)

(c)

Fig. 2. (a) The benefits hierarchy. (b) The costs hierarchy. (c) The risks hierarchy.


a time. When all issues are considered jointly, its column is named ‘‘Overall Rating’’ (OR). Figs. 5b and care derived similarly.

The local priority, plijk, can be converted to global priority, gl

ijk, by weighting the corresponding issue’spriority, wijk. For example, the local priorities for each of the three (reform, no reform) pairs under the

ARTICLE IN PRESS

(a)

Benefits Employment Investment Infrastructure Societal Harmony Weight (wij)

EmploymentInvestmentInfrastructureSocietal Harmony

1 3 1 5

1/3 1 1 2

1 1 1 3

1/51/21/31

0.110 (=c11) 0.238 (=c12) 0.164 (=c13) 0.488 (=c14)

Costs Deficit Unfairness Less Effic. Gov. Less Op. & Ch. Weight

Deficit Unfairness Less Efficient Government Less Opportunity& Choice

1 3

1

2

1/3 1

1/3

1/2

1 3

1

1/2

1/2 2

2

1

0.143 (=c21) 0.452 (=c22)

0.212 (=c23)

0.193 (=c24)

Risks Weak Compet. Class War Burden Weight

Weak CompetitionClass War Burden Future Generation

1 1/21/3

2

1/2

3 2 1

0.540 (=c31) 0.297 (=c32) 0.163 (=c33)

(b)

Step 1: Sum the values in each columnBenefits Em

Em

Em

ployment Investment Infrastructure Societal Harmony

EmploymentInvestmentInfrastructureSocietal Harmony Column Sum

1 3 1 510

1/31 1 213/3

1 1 1 36

1/51/21/3161/30

Step 2: Divide each element by its column sumBenefits ployment Investment Infrastructure Societal Harmony


1/103/101/105/10

1/133/133/136/13

1/61/61/63/6

6/6115/6110/6130/61

Step 3: Average the elements in each row Benefits ployment Investment Infrastructure Societal

HarmonyWeight


0.100 0.300 0.100 0.500

0.077 0.231 0.231 0.462

0.1670.1670.1670.500

0.0980.2460.1640.492

0.110 (=c11) 0.238 (=c12) 0.164 (=c13) 0.488 (=c14)

Fig. 3. Deriving the weights for factors. (a) Pairwise comparisons in each hierarchy. (b) Approximating the weights of the benefit factors.


ARTICLE IN PRESS

Benefits Employment Tax

ReformAA Reform

TermLimit

Scale Investment Tax Reform

AA Reform

TermLimit

Scale

Tax ReformAA ReformTerm Limit

1 1/71/8

7 1 1/3

8 3 1

0.777 (=w111) 0.153 (=w112) 0.070 (=w113)


1 1/8 1/6

8 1 1/2

6 2 1

0.773 (=w121) 0.134 (=w122) 0.093 (=w123)

Infrastructure TaxReform

AA Reform

TermLimit

Scale Societal Harmony TaxReform

AA Reform

Term Limit Scale


1 1/41/5

4 1 2

5 1/2 1

0.687(=w131)0.127(=w132)0.186(=w133)


1 1/4 1/3

4 1 3

3 1/3 1

0.614 (=w141) 0.117 (=w142) 0.268 (=w143)

Costs Deficit

ReformAA Reform

Term Limit Scale Unfairness TaxReform

AAReform

Term Limit Scale


1 1/9 1/8

9 1 1/2

8 2 1

0.806 0.117 0.077


1 4 1/4

1/4 1 1/6

4 6 1

0.236 0.682 0.082

LessGovernment

TaxReform

AA Reform

Term Limit Scale Less Opportunity & Choice

TaxReform

AAReform

Term Limit Scale


1 1 5

1 1 4

1/5 1/4 1

0.149 0.160 0.691


1 3 4

1/3 1 2

1/4 1/2 1

0.122 0.320 0.558

RisksWeakCompetitiveness

TaxReform

AA Reform

TermLimit

Scale Class War TaxReform

AA Reform TermLimit

Scale


1 2 1

1/21 1

1 1 1

0.260 0.413 0.327


1 2 1/3

1/21 1/6

3 6 1

0.3000.6000.100

Burden Tax Reform AA Reform Term Limit Scale

Tax Reform

AA Reform

Term Limit

1

1/8

1/8

8

1

1/4

8

4

1

0.783

0.155

0..62

Tax

Efficient

Fig. 4. Pairwise comparison matrices for all issues. Note: AA reform stands for affirmative action reform.


increase employment criterion are: (p1111, p2

111) ¼ (0.875, 0.125), (p1112, p2

112) ¼ (0.667, 0.333), and (p1113,

p2113) ¼ (0.100, 0.900). Weighted by wijk, the global priority becomes (g1

111, g2111) ¼ (0.680, 0.097), (g1

112,

g2112) ¼ (0.102, 0.051), and (g1

113, g2113) ¼ (0.007, 0.063). By applying step 2(d) of Fig. 1, we obtain the ORs

for each alternative as shown in the rightmost column of Figs. 5a–c. These numbers are then used todevelop the final ratios in Table 2.

(iii)
Deriving the ratio.
Because we respond to the question ‘‘which is more costly and which is more risky?’’ when combining the
priorities derived from the three hierarchies to obtain the final outcome we can use, as appropriate, themarginal or the total method of aggregating benefits, costs, and risks. For simplicity, we only illustrate withthe simpler marginal formula as it does not require weighting these three different merits of the alternatives.[25, p. 89, 120]. Thus for our example, we divide the benefits results from the benefits hierarchy by those fromthe costs and risks hierarchies (see Table 2). In our example, we find that the benefits of tax reform are thehighest among all alternatives, while its corresponding costs and risks are a little greater than those of the

ARTICLE IN PRESS

IncreaseEmployment

0.110

Stimulate Investment & Economy0.238

StrengthenTechnology &Infrastructure 0.164

Benefits

PromoteSocietal Harmony0.488

LocalRating0.6340.366

OverallRating0.4480.233

Local0.8750.125

Local0.8750.125

Global0.6800.097

Global0.7050.101

Local0.8890.111

Global0.6870.086

Local0.8000.200

Global0.5500.137

Local0.4000.600

Global0.2460.368

Tax Reform No Tax Reform

AA Reform 0.667 0.102 0.400 0.054 0.250 0.032 0.100 0.012 0.258 0.035 No AA Reform 0.333 0.051 0.600 0.080 0.750 0.095 0.900 0.105 0.742 0.092

Term Limit 0.100 0.007 0.333 0.031 0.500 0.930 0.600 0.161 0.465 0.102 No Term Limit 0.900 0.063 0.667 0.062 0.500 0.930 0.400 0.107 0.535 0.089

LR121

OR132

P1122 P123

1

DeepenDeficit

0.143

Unfairness Perception

0.452

LessEffective

Government 0.212

Costs

Less Opportunities

& Choices 0.193



Local0.4000.600

Global0.0940.142

Local0.3330.667

Global Local0.3330.667

Global0.0410.081

Tax Reform .050 No Tax Reform .099

Affirmative Action Reform0.500 0.059 0.750 0.512 0.667 .107 0.800 0.256 0.706 0.312 No Affirmative Action

Reform 0.500 0.059 0.250 0.171 0.333 .053 0.200 0.064 0.294 0.109

Term Limit 0.500 0.039 0.143 0.012 0.900 .622 0.800 0.446 0.481 0.229 No Term Limit 0.500 0.039 0.857 0.070 0.100 .069 0.200 0.112 0.519 0.073

Long TermNegative

Competitiveness.540

Class War

0.297

Unfair Burdenon Future

Generation 0.163

Risks



Local0.2000.800

Global0.0520.208

Local0.6000.400

Global0.1800.120

Local0.8570.143

Global0.6710.112

Tax ReformNo Tax Reform

Affirmative Action Reform 0.600 0.248 0.800 0.480 0.500 0.078 0.643 0.289

No Affirmative Action Reform 0.400 0.165 0.200 0.120 0.500 0.078 0.357 0.137

Term Limit 0.400 0.131 0.400 0.040 0.333 0.021 0.389 0.086No Term Limit 0.600 0.196 0.600 0.060 0.667 0.042 0.611 0.130

(a)

(b)

(c)

Fig. 5. Global rating for each alternative: (a) benefits hierarchy; (b) costs hierarchy; (c) risks hierarchy.


ARTICLE IN PRESS

Table 2

Deriving ratios for alternatives

Calculation of benefits/(costs� risks) ratio To do or not to do

Tax Reform0:448

0:162� 0:191¼ 14:48

Yes (Tax Reform has higher ratio than No Reform)

No Tax Reform0:233

0:115� 0:166¼ 12:205

Affirmative Action Reform0:035

0:312� 0:289¼ 0:3882

No (No Affirmative Action Reform far exceeds Reform)

No Affirmative Action Reform0:092

0:109=0:137¼ 6:161

Term Limits0:102

0:229� 0:086¼ 5:1791

No (No Term Limits dominates Term Limits)

No Term Limits0:089

0:073� 0:130¼ 9:3782


no tax reform alternative. Therefore, the tax reform ratio is higher than that of the no tax reform. The formerdominates the latter both when no risk is considered, and also when projected risk is taken into account.

Compared with affirmative action reform, the no affirmative action reform benefits are higher, with costs andrisks lower. Thus, the no affirmative action reform overall ratio is much higher than that of the affirmative

action reform decision.Term limits dominates no term limits when costs are not considered. When costs are taken into account,

however, no term limits has a higher priority than term limits. We note that including risks by using possiblescenarios of the future can be a powerful tool in assessing a decision. The same procedure can be applied if onechooses a different merit system. For example, instead of using benefits, opportunities, costs, and risks for theanalysis framework, one may choose strength, weakness, opportunity, and threat (SWOT) to evaluate thealternatives. We find that the proposed approach is useful in either framework.

3.2.3. Linked issues

If we encounter a situation in which acting on one of the issues requires that we also act on another so thattwo issues appear together, then we would consider the pair as a single issue. The sum of the B/CR ratios (i.e.,Benefits/(Costs�Risks)) for changing the two may exceed the sum of not doing both, even though takensingly, one of them may be rejected. For example, if term limits is required for tax reform, we should add theB/CR ratio of both (14.48+5.179 ¼ 19.659) and compare it with the sum of no tax reform and no term limits

(12.205+9.378 ¼ 21.5834). Since the latter is larger, we would carry out neither tax reform nor term limits.The object of this process is to integrate the issues so that decision-makers would not arbitrarily decide oneach issue alone as they do in ordinary voting.

By linking an issue’s B/CR ratio to the ratios of the other issues, the decision-makers can weight it carefullyby assigning the appropriate strength (judgment). Otherwise, exaggerating its value would, in theproportionality and normalization scheme, unduly reduce some other issue of its desired priority. Thisappears to be a compelling method for determining the relative importance of issues rather than simply givingone of them too high a value and the other a correspondingly very low one, or just voting yes–no the issues.

3.2.4. Sensitivity analysis

To ensure that the outcome of the above example not be construed as a result of whimsical judgments, weperformed a comprehensive sensitivity analysis on the 11 main criteria (four under the benefits hierarchy, fourunder the costs hierarchy, and three under the risks hierarchy). A sensitivity analysis helps determine therobustness of a model. It tests a plausible range of values for each criterion to determine how sensitiveoutcomes are to changes in the input estimates. Through sensitivity analysis, policy-makers can discover howchanges in judgments or priority about the importance of each criterion might affect recommended decisions.We change the importance of each criterion by up to 730%. That is, we tested each criterion by varying itsoriginal priority value to within (70%, 130%) range.


In the meantime, the minimum priority value is limited to 0 and the maximum to 1. Since each criterion isallowed to vary twice, and we have four criteria for benefits, in all we would have 2� 4 ¼ 8 variations in thebenefits hierarchy. Similarly there would be 2� 4 ¼ 8 variations in the costs and 2� 3 ¼ 6 variations in therisks hierarchies. To cover all possible interactions, we generated 8� 8� 6 ¼ 384 data points. We found thattax reform dominates no tax reform 95.1% of the time. When the burden on future generations and class warrisks are considered much more important, no tax reform dominates. No affirmative action reform dominatesaffirmative action reform 92.9% of the time, and no term limits is preferred to term limits in about 54.8% of thecases. The results suggest that tax reform is the most pressing issue, while affirmative action reform is notpreferred at this time. The implications of term limits are not particularly decisive in the debate due to its lowpercentage in dominance.

The public policy-making example given above takes into account preference intensity, so that collectivepriority is not determined merely by head-counting. It accommodates both objective and subjective judgment,and individual shared values, thus allowing all pertinent informations to be deliberated. Users can input andmodify the data format, and synthesize the results. More importantly, the example considers both winners’and losers’ positions. Therefore, the proposed approach can contribute to building consensus and accord.

4. Internet and group decision-making

In this section, we examine two potential useful applications of our proposed framework using computersand the Internet. The first is in public policy voting, and the second in business group decision-making.

4.1. Public policy-making through Internet voting

The idea of using Internet technology to facilitate the voting process has engendered much interest in oursociety, particularly after the technical glitches in the US presidential election of 2000 [26]. E-voting (electronicvoting, Internet voting or online voting) is a method that transmits voters’ choices over the Internet throughsecure encryption. There are three Internet electronic voting models: (i) electronic voting at conventional sites,where traditional voting locations are enhanced with Internet technology; (ii) remote voting, where voters votefrom home or work; and (iii) kiosk voting, where Internet terminals are placed at convenient sites such asmalls. Remote voting holds the greatest promise due to its convenience and universal access.

In reforming and modernizing the conventional voting system, many have studied the social and technicalimplications of e-voting [27]. Problems such as how e-voting affects voters’ presence, and if different sectors ofthe population participate differently have drawn much attention [28]. An important concern is that those whohave no Internet access from home or work are digitally divided from those who do, and thus they becomedisadvantaged. In addition to the lack of equal Web access, critics also worry about Internet security. Sincecomputers and the Internet are fundamentally vulnerable, the threat of an election ruined by malicious attack,software glitches and mechanical errors is real and daunting. It is important to protect privacy and safeguardagainst ‘‘hackers’’ so those with culpable intent cannot disrupt the casting and tallying of the votes through theInternet.

In governmental elections, the need to create a secure process for collecting and counting votes must beaddressed satisfactorily before online voting becomes a viable option. Other concerns include verifyingaccuracy and authenticating voters (is this the only ballot that the voter sent? Is the voter the person that sentthis ballot?). Digital signature technology may help accomplish such a verification function. While this type oftechnology can protect privacy and secrecy, its cost remains high.

Public policy-making has, thus far, not taken full advantage of current information technology (IT). Withthe popularity of the Internet, our framework may well be implemented on the Internet to bring the generalpublic and government closer together. Such implementation could permit wider participation and morecareful deliberation within the democratic process than in conventional voting systems. The Internet PolicyInstitute [29,30] has been promoting government use of the Web, and is working toward developing aneffective e-voting system. However, developments thus far have revolved around the fundamental yes–no typeof voting, using the traditionally familiar 0–1 concept in the digital era.


The yes–no method is popular mainly because no convenient way exists to express preference intensities onballot paper. It is true that putting marks on a ballot is simple and easy to count, whereas synthesizingintensity of preferences is not as straightforward. However, the advance of computer technology and theemergence of software have changed such concerns. Through prudent analysis and design, and improvingInternet technology in safety, security, accessibility, integrity, and cost, a wide-ranging Internet application forpublic voting is foreseeable.

One may safely assume that policies are more likely to be implemented successfully when they are thoughtthrough carefully beforehand. Through an Internet implementation of our AHP framework, voters would beable to send their preferences using computers, so that individual preferences can be heard, and popularsentiments and choices could be brought together. This promises to be a more proactive and participatoryprocess, and will likely lead to better communication and more efficient governance.

4.2. Web-enabled framework for business application

In addition to governmental elections, businesses and small groups may also benefit from Internet votingand Web-based group decision support systems (DSS). In today’s world, every organization should have at itsdisposal an efficient group decision-making method that permits its members to express their preferencesthrough computers. Since the Internet has become an integral part of our daily life, it offers a valuableopportunity for collecting opinions.

The decision support framework we proposed above can easily be Web-enabled to allow employees andstakeholders to participate through Internet access. Our approach may, for example, be useful for a smallmanagement team allocating a company’s funds or for a 600-person marketing force seeking to contributeideas to new product design. For example, in Ref. [31], it is reported that the AHP-based multi-criteria groupdecision-making approach has been used by Lockheed Martin to select new IT products and services; by IBMto benchmark industry competition and to better allocate resources; by the Federal Aviation Agency (FAA) toprioritize R&D projects; by Housing and Urban Development (HUD) to manage their IT portfolio andrequest funding from Congress; and by the US Army to plan multi-year budgets. Ford has also employed anAHP-based DSS to assess customer satisfaction and to evaluate next-generation vehicle designs [32]. Theseapplications can be adapted for the Internet, allowing for greater and more effective participation in decision-making. By allowing users to give critical input from anywhere in the world, the software may eliminate theneed for some business travel and face-to-face meetings.

Our web-based DSS can also allow users to set priorities on criteria that are linked to remote databases. Forexample, it can help establish criteria for selecting suppliers by linking to each supplier’s database with pastperformance characteristics. The decision-maker first sets priorities on performance characteristics relevant toprocuring supplies. The system then links to each remote supplier’s database and brings supplier data directlyto the model. The Internet can thus bring greater value to the relationship between priorities and remote datain decision.

5. Conclusions

In this research, we have identified some key shortcomings of traditional majority voting, in which publicpolicy-makers have little or no way to express the intensity of their preferences. We thus developed a votingprocedure that is objective, takes into consideration each individual’s sentiments, and allows for reasoningbased on a structured evaluation framework. The approach is general and relatively easy to understand, andputs voting in a richer human context to match more closely how people feel about issues. It could have wideapplicability in the real world, particularly in business and politics where one needs to know the best outcomesof debate and negotiation. The proposed ‘‘cardinal’’ approach helps ensure that one thinks things throughso that he/she can provide fair judgments to derive priorities.

We believe that a voter’s true feelings can be elicited because people are generally concerned with (1) moral(conscience) obligation, (2) law requirements, (3) personal credibility, and (4) averting social chaos(if everyone misleads others, society will not achieve harmony). When multiple issues are considered, theproposed AHP method prohibits one from overstating the importance of an issue without sacrificing the


priority of the other issues. In life, in business, and in all political decisions, various criteria and multiple issuesare often contemplated simultaneously; yet, single issues are regularly considered with their correspondingnegations. A comprehensive evaluation approach helps one create more value for the quality of life, for theprofit of a company, or for the well-being of society. A useful approach for dealing with factor dependenceand feedback matters within the B/CR ratio framework is the Analytic Network Process (ANP) [25]. Due tothe limitation of this paper’s scope, we will leave this consideration for future study.

We assume that the AHP is a practical, analytical tool to use. We also understand that the world will nothasten to implement our proposal. Rather, we believe this method, which includes how strongly people feelabout their choices (an integral part of their biological and psychological makeup), provides a method formaking policies and decisions that is more precise and valid than earlier approaches.

Our method is also sensible in that it explores strength of preference across issues. A limitation is that theprocedure does not fully explore the relative strengths and weaknesses of alternative methods. However, suchcomparisons have been made by Peniwati [33].

Possible strengths of the proposed voting approach include the following:

(1)
It provides a structured decision-making approach and aligns with organizational and strategic goals. (2) It brings about conditions conducive to better, faster, more justifiable decisions, and helps generate
consensus, improve communication, and thus enhance results.
(3) It allows groups to weigh team members’ preferences and evaluate outcomes based on demographics,
while providing a capability for sensitivity analysis.
(4) Through IT, it may accept judgments from multiple stakeholders using wireless keypads or the Internet for
same-time, different-time, same-place, and/or remote decision-making.

Some weaknesses of our proposed method may be:

(1)
It is difficult to change the current voting system. (2) It would take time to implement the approach. (3) People may be reluctant to use it because they have to justify their own preferences, rather than simply
saying yes or no.
(4) It requires implementing a new way of making decisions.
The concept presented in this paper is preliminary; we hope that the proposal will help draw furtherattention to the subject.

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Thomas L. Saaty obtained his Ph.D. in mathematics from Yale University. He holds the Chair of University Professor, Katz Graduate

School of Business, University of Pittsburgh, Pittsburgh, PA. He was previously Professor, Wharton School of Business, University of

Pennsylvania. Professor Saaty spent seven years at the Arms Control and Disarmament Agency in the US State Department, during which

time major arms reduction negotiations were held with the Soviets in Geneva. His current research interests include decision-making,

planning, conflict resolution, and synthesis of signals in the brain. As a result of his search for an effective means to deal with weapons

tradeoffs at the Disarmament Agency and, more generally, with decision-making and resource allocation, Professor Saaty developed The

Analytic Hierarchy Process (AHP) and its generalization to dependence and feedback, the Analytic Network Process (ANP). He is co-

developer of the software Expert Choice and of the software Super Decisions for decisions with dependence and feedback. He has

authored and co-authored more than a dozen books on the AHP/ANP. Professor Saaty has also written a number of other books that

embrace a variety of topics, including Modern Nonlinear Equations, Nonlinear Mathematics, Graph Theory, The Four Color Problem,

Behavioral Mathematics, Queuing Theory, Optimization in Integers, and Embracing the Future and The Brain: Unraveling the Mystery of

How It Works. His most recent book is Creative Thinking, Problem Solving & Decision Making. The book is a rich collection of ideas,

incorporating research by a growing body of researchers and practitioners, profiles of creative people, projects and products, theory,

philosophy, physics and metaphysicsyall explained with a liberal dose of humor. He has published more than 300 refereed articles in a

wide variety of professional journals. He has been on the editorial boards of Mathematical Reviews, Operations Research, Naval Research

Logistics Quarterly, Mathematical and Computer Modeling, Socio-Economic Planning Sciences, Applied Mathematics Letters, and several

others. He also served as a consultant to many corporations and governments. In 2005, he was elected to the US National Academy of

Engineering.

Jennifer Shang is Associate Professor, Katz Graduate School of Business, University of Pittsburgh. She received her Ph.D. in Operations

Management from the University of Texas at Austin. Her main research interests include multi-criteria decision-making and its

application to the design, planning, scheduling, control, and evaluation of production and service operational systems. She has published

in various journals, including Management Science, European Journal of Operational Research, IEEE Transactions on Engineering

Management, and International Journal of Production Research. She has won the 2005 EMBA Distinguished Teaching Award, and several

Excellence in Teaching awards from the MBA/EMBA programs at Katz.

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group decision-making: head-count versus intensity of preference

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